Advancement: A Nonparametric Bayesian Modeling Framework for Flexible Ordinal Regression

Jizhou Kang
Statistical Science PhD Student
Location
Virtual Event
Advisor
Athanasios Kottas

Join us on Zoom: https://ucsc.zoom.us/j/6644523718?pwd=NVZTYTByQURlanN1NmxYeG55V0JZdz09 / Passcode: 860125

Description: Ordinal responses measured along with covariates are present in a variety of problems, under either the cross-sectional or longitudinal setting. Traditional parametric approaches to ordinal regression are typically based on the assumption that the ordinal responses arise through discretization of a latent continuous distribution, with covariate effects entering linearly. While they enjoy simplified inference, limitations in the covariate-response relationship and/or the underlying response distribution hinder their applications. The demand for flexible and easily implemented models which relax the common restrictions is clear. Through the use of Bayesian nonparametric modeling techniques, nonstandard features of regression relationships may be obtained if the data suggest them to be present. We develop Bayesian nonparametric modeling approaches to ordinal regression based on priors placed directly on the discrete distribution of the ordinal responses. The prior probability models are built from a structured mixture of multinomial distributions. We leverage a continuation-ratio logits representation and Pólya-Gamma augmentation to guarantee a computationally efficient posterior simulation method. We start by building models for cross-sectional ordinal regression. The general nonparametric mixture model has the logit stick-breaking process (LSBP) structure on both the weights and the atoms. The implied regression functions for the response probabilities can be expressed as weighted sums of regression functions under traditional parametric models, with covariate-dependent weights. Thus, the modeling approach achieves flexibility in ordinal regression relationships, avoiding linearity or additivity assumptions in the covariate effects. Moving towards modeling the dynamic ordinal regression relationship from longitudinal studies, we first consider semiparametric models using hierarchical Gaussian process (GP) priors. Next, combining the hierarchical GP and LSBP prior structures, we propose nonparametric modeling methods, including extensions to incorporate covariate effects. The proposed models are illustrated with several synthetic and real data examples. Finally, we tailor these models to two applications, namely estimating dose-response curves in developmental toxicity studies, and estimating fish maturity under different experimental conditions.